On the Chaotic Representation Property of Certain Families of Martingales and Applications
H.- J. Engelbert

Резюме:
We start by giving a brief historical review of what is known about the chaotic representation property (CRP) in terms of normal random measures and independent random measures. In this context, there arises the natural question whether the CRP can be extended to more general orthogonal random measures. Thereafter we move on to families X of square integrable martingales, the main object of our investigation. Under the assumption that their predictable covariations are deterministic, we shall introduce iterated integrals and state their properties. On this basis, the definition of the CRP will be given. An important property of X is its so-called compensated-covariation stability. It turns out that under this additional condition on X, every stochastic integral of monomials in martingales from X with respect to martingales from X again belongs to the space of elementary iterated integrals J_e. This seems an important property, one of the main ingredients for the further developments. We continue by providing sufficient conditions on X for possessing the CRP. As a first illustration of this general result, then we discuss Gaussian families of (local) martingales and independent families of compensated Poisson processes. Thereafter we focus our attention to Levy processes L: We consider certain families X of square integrable martingales with respect to the Levy filtration F^L and we provide necessary and suffficient conditions for the CRP of X. As a special case, under certain conditions on the Levy measure of L, this includes the so-called Teugels martingales studied by Nualart & Schoutens (2000). We close by discussing some more examples and potential applications of the stated results on the CRP.

27, April 2016
12:00 Wednesday

On the Chaotic Representation Property of Certain Families of Martingales and Applications
H.- J. Engelbert

Резюме:
We start by giving a brief historical review of what is known about the chaotic representation property (CRP) in terms of normal random measures and independent random measures. In this context, there arises the natural question whether the CRP can be extended to more general orthogonal random measures. Thereafter we move on to families X of square integrable martingales, the main object of our investigation. Under the assumption that their predictable covariations are deterministic, we shall introduce iterated integrals and state their properties. On this basis, the definition of the CRP will be given. An important property of X is its so-called compensated-covariation stability. It turns out that under this additional condition on X, every stochastic integral of monomials in martingales from X with respect to martingales from X again belongs to the space of elementary iterated integrals J_e. This seems an important property, one of the main ingredients for the further developments. We continue by providing sufficient conditions on X for possessing the CRP. As a first illustration of this general result, then we discuss Gaussian families of (local) martingales and independent families of compensated Poisson processes. Thereafter we focus our attention to Levy processes L: We consider certain families X of square integrable martingales with respect to the Levy filtration F^L and we provide necessary and suffficient conditions for the CRP of X. As a special case, under certain conditions on the Levy measure of L, this includes the so-called Teugels martingales studied by Nualart & Schoutens (2000). We close by discussing some more examples and potential applications of the stated results on the CRP.

26, April 2016
12:00 Tuesday

On the Chaotic Representation Property of Certain Families of Martingales and Applications
H.- J. Engelbert

Резюме:
We start by giving a brief historical review of what is known about the chaotic representation property (CRP) in terms of normal random measures and independent random measures. In this context, there arises the natural question whether the CRP can be extended to more general orthogonal random measures. Thereafter we move on to families X of square integrable martingales, the main object of our investigation. Under the assumption that their predictable covariations are deterministic, we shall introduce iterated integrals and state their properties. On this basis, the definition of the CRP will be given. An important property of X is its so-called compensated-covariation stability. It turns out that under this additional condition on X, every stochastic integral of monomials in martingales from X with respect to martingales from X again belongs to the space of elementary iterated integrals J_e. This seems an important property, one of the main ingredients for the further developments. We continue by providing sufficient conditions on X for possessing the CRP. As a first illustration of this general result, then we discuss Gaussian families of (local) martingales and independent families of compensated Poisson processes. Thereafter we focus our attention to Levy processes L: We consider certain families X of square integrable martingales with respect to the Levy filtration F^L and we provide necessary and suffficient conditions for the CRP of X. As a special case, under certain conditions on the Levy measure of L, this includes the so-called Teugels martingales studied by Nualart & Schoutens (2000). We close by discussing some more examples and potential applications of the stated results on the CRP.

Резюме:
Microsatellites are defined by several repetitions of the same
coding sequence. The typical mutation will add or subtract some
repetitions. We will present a mathematical (spatial) model inspired by Wright-Fisher
and adding distances of mutations. These models are in the
mathematical framework of iterated functions systems and coalescent.
The global average length of microsatellites within a population
behaves in time like a random walk. The local view of the length
shows a tendency to be close to the average. The distribution
of the distance is a stationary measure of a certain Markov chain.
This macroscopic and microscopic phenomenon is observable in real data.